Given the latitudes and longitudes of two places on earth, what is the (great circle) distance between them?

• The earth is (generally close enough to) a sphere, radius R; define the (surface) distance we seek as D.
• D=Ra where a is the angle (in radians, or in degrees*(2Pi/360)) between vectors R₁ and R₂ from the center of the earth to the two points.
• Since the dot product R₁.R₂=R²cosa, we can write D=Rcos^-1(R₁.R₂/R²)
• Transform spherical to rectilinear coordinates: (R,lat,long)->(Rcos(lat)cos(long),Rcos(lat)sin(long),Rsin(lat)).
• So R₁.R₂/R²=cos(lat₁)cos(long₁)cos(lat₂)cos(long₂)+cos(lat₁)sin(long₁)cos(lat₂)sin(long₂)+sin(lat₁)sin(lat₂).
• D=R cos^-1(cos(lat₁)cos(long₁)cos(lat₂)cos(long₂)+cos(lat₁)sin(long₁)cos(lat₂)sin(long₂)+sin(lat₁)sin(lat₂)).
• D=R cos^-1(cos(long₁-long₂)cos(lat₁)cos(lat₂)+sin(lat₁)sin(lat₂)).
• Sanity check: On the equator (both latitudes zero) this says D=R(long₁-long₂) as expected.
• Sanity check: When the two longitudes are equal this says D=R(lat₁-lat₂) as expected.